This is the third paper of a series devoted to the study of the global properties of Joguet's sample of 79 nearby galaxies observable from the southern hemisphere, of which 65 are Seyfert 2 galaxies. We use the population synthesis models of Paper II to derive ‘pure’ emission-line spectra for the Seyfert 2 galaxies in the sample, and thus explore the statistical properties of the nuclear nebular components and their relation to the stellar populations. We find that the emission-line clouds suffer substantially more extinction than the starlight, and we confirm the correlations between stellar and nebular velocity dispersions and between emission-line luminosity and velocity dispersions, although with substantial scatter. Nuclear luminosities correlate with stellar velocity dispersions, but Seyferts with conspicuous star-forming activity deviate systematically towards higher luminosities. Removing the contribution of young stars to the optical continuum produces a tighter and steeper relation, L∝σ4★, consistent with the Faber-Jackson law. Emission-line ratios indicative of the gas excitation such as [O iii]/Hβ and [O iii]/[O ii] are statistically smaller for Seyferts with significant star formation, implying that ionization by massive stars is responsible for a substantial and sometimes even a dominant fraction of the Hβ and [O ii] fluxes. We use our models to constrain the maximum fraction of the ionizing power that can be generated by a hidden active galactic nucleus (AGN). We correlate this fraction with classical indicators of AGN photoionization (i.e. X-ray luminosity and nebular excitation), but find no significant correlations. Thus, while there is a strong contribution of starbursts to the excitation of the nuclear nebular emission in low-luminosity Seyferts, the contribution of the hidden AGN remains elusive even in hard X-rays.

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